• Artin groups
• discrete Morse theory

The aim of this thesis is to present the K(pi,1) conjecture for Artin groups, an open conjecture which goes back to the 70s, and to use the technique of discrete Morse theory to prove some results connected with it.

The beginning of the study of Artin groups dates back to the introduction of braid groups in the 20s.
Artin groups where defined in general by Tits and Brieskorn in the 60s, in relation with the theory of Coxeter groups and singularity theory.
Deep connections with the main areas of mathematics where discovered: in addition to the theory of Coxeter groups and singularity theory, they naturally arise in the study of root systems, hyperplane arrangements, configuration spaces, combinatorics, geometric group theory (see the very recent solution of the virtually Hacken conjecture), knot theory, mapping class groups and moduli spaces of curves.

There are many properties conjectured to be true for all Artin groups but proved only for some families of them, e.g. being torsion-free, having a trivial center, and having solvable word problem.
Some of these problems, and also others (such as the computation of homology and cohomology), are related to an important conjecture called "K(pi,1) conjecture".
Such conjecture says that a certain topological space N, constructed in a certain way from a fixed Coxeter group, is a classifying space for the corresponding Artin group.
The space N admits finite CW models, therefore the K(pi,1) conjecture directly implies that Artin groups are torsion-free.

A tool which is very important in our work is discrete Morse theory, introduced by Forman in the 90s.
Discrete Morse theory allows to prove the homotopy equivalence of CW-complexes through elementary collapses of cells, on the basis of some combinatorial rules which can be naturally expressed with the language of graph theory.
The idea of using discrete Morse theory to prove results about the K(pi,1) conjecture is present in the literature only in very recent works.

This thesis is structured as follows.
In the first chapter we present some of the most important known results about Coxeter groups, especially concerning their geometric and combinatorial properties.
In the second chapter we do the same for Artin groups. In particular we introduce the Artin monoids, which are significantly important in the study of Artin groups.
The third chapter is devoted to an introduction to the terminology and the main results of discrete Morse theory, in a version developed by Chari and Batzies after the original work of Forman.
In the fourth chapter we introduce the K(pi,1) conjecture together with some of its consequences.
We define a particular CW model for the space N, called Salvetti complex, and we describe its combinatorial structure.
Then we give a new proof of the K(pi,1) conjecture for Artin groups of finite type (i.e. those for which the corresponding Coxeter group is finite), using discrete Morse theory.
Finally, in the fifth chapter we describe some connections between the K(pi,1) conjecture and classifying space of Artin monoids. A relevant result in this direction is a theorem by Dobrinskaya published in 2006, which states that the classifying space of an Artin monoid is homotopy equivalent to the corresponding space N mentioned above.
We prove that applying discrete Morse theory one can collapse the standard CW model for the classifying space of an Artin monoid and obtain the Salvetti complex. In particular, this gives an alternative prove of Dobrinskaya's theorem.